UNIVERSIT ` A DEGLI STUDI DI PADOVA
DIPARTIMENTO DI FISICA E ASTRONOMIA “GALILEO GALILEI”
DIPARTIMENTO DI INGEGNERIA DELL’INFORMAZIONE Corso di Laurea Magistrale in Fisica
TESI DI LAUREA MAGISTRALE
Violation of a tripartite Bell inequality by weak measurements
Relatore: Dott. Giuseppe Vallone Laureando: Mirko Pittaluga Controrelatore: Dott. Michele Merano Matricola: 1104727
Anno Accademico 2015 - 2016
Dedicato ai miei genitori e a mia sorella senza i quali questa avventura non sarebbe stata possibile.
A B S T R A C T
Non-locality is one of the most characteristic trait of Quantum Mechanics, and it is linked to the conceptually counter-intuitive ability of some objects (when adequately prepared) to instantaneously know about each other’s state, no matter how much space-separated they are. At the development stage of Quantum Mechanics, non- locality had been an argument of intense debate among scientists [1] [2] due to its consequences that seemed to contradict physicist common sense. The scientific debate on the argument faced a turning point with the publication of the landmark paper by J. S. Bell in 1964 [3] which ended the discussion whether or not non-locality was a true aspect of physical reality, raising it to one of the fundamental aspect of our comprehension of the physical world.
The scientific interest for non-locality has recently substantially increased due to its striking applications in Quantum Information Theory [4] [5] [6]. The major innovative traits of Quantum Information Theory, compared to Classical Information Theory, arise in fact from the deployment of non-locality as a resource for information manipulation and transmission.
The aim of this thesis is to shed a new light on the argument from an experimental point of view, proving whether is possible or not to share non-local correlations among three observers sharing an entangled two photon pair. This work is one of the first to address this issue, and aims to prove experimentally what has been hypothesized theoretically by Silva, et al. in their 2015 article [7].
A C K N O W L E D G E M E N T
I would like to express my deepest gratitude to Matteo Schiavon e Luca Calderaro, the two Ph.D. students of Quantum Future research group I have joined in this project.
From the experiment design to the data analysis, their assistance and suggestions have accompanied me through all the learning process of this master thesis.
I would also like to thank my supervisor Dr. Giuseppe Vallone for his guidance, supervision and example. It is probably due to his support and his advices that this project could end up successfully.
Finally I would like to thank all the members of Quantum Future research group in Padua, who made my the time spent on this thesis such an enjoyable, stimulating and productive experience.
C O N T E N T S
Cover Page i
Abstract v
Acknowledgement vii
Table of Contents ix
List of figures xi
1 i n t r o d u c t i o n 1
2 n o n-locality and weak measurement 5
2.1 Qubit . . . 5
2.1.1 Multiple qubit systems . . . 7
2.2 Entangled states and EPR paradox . . . 7
2.3 Non-locality, Bell Inequalities and CHSH inequality . . . 8
2.3.1 Mathematical characterization of non-local correlations . . . . 11
2.4 Weak measurement . . . 14
2.4.1 Modelling the ancilla measurement process . . . 15
2.4.2 The pre-measurement . . . 15
2.4.3 The read-out . . . 17
3 n o n-locality sharing among multiple observers 19 3.1 Tripartite Bell Inequality . . . 19
4 e x p e r i m e n ta l t e c h n i q u e s 23 4.1 Spontaneous parametric down-conversion . . . 23
4.2 Experimental design of the source . . . 28
5 e x p e r i m e n t 31 5.1 Theoretical model . . . 31
5.1.1 Trivial circuit example . . . 32
5.1.2 System circuit . . . 32
5.1.3 State evolution . . . 34
5.1.4 State measurement and probability . . . 36
5.1.5 CHSH correlations and inequalities . . . 39
5.2 Experimental design . . . 43
5.2.1 Experimental scheme . . . 43
5.2.2 Equivalence between experiment design and theoretical model 47 5.3 Apparatus . . . 48
5.3.1 Apparatus description . . . 50
6 e x p e r i m e n ta l r e s u lt s 55
Contents
6.1 Apparatus characterization . . . 55
6.1.1 Glass Slide Characterization . . . 55
6.1.2 Half Wave Plate Characterization . . . 57
6.1.3 Polarization shift characterization . . . 59
6.1.4 System stability . . . 60
6.2 State preparation procedure . . . 61
6.2.1 Source alignment . . . 64
6.2.2 Interferometer optimization . . . 65
6.3 Experiment procedure . . . 66
6.4 Experiment results . . . 66
6.4.1 CHSH correlation values VS . . . 67
6.4.2 Double CHSH inequality violation . . . 68
7 c o n c l u s i o n s 71 Appendices 73 a q ua n t u m i n f o r m at i o n t h e o r y 75 a.1 Quantum mechanics . . . 75
a.2 The density matrix formalism . . . 76
a.2.1 Subsystems and purification . . . 77
a.2.2 Generalized measurements . . . 78
a.3 The circuit model . . . 78
a.3.1 Quantum wires . . . 79
a.3.2 Quantum gates . . . 79
a.3.3 Measurement . . . 81
b e l e m e n t s o f c l a s s i c o p t i c s 83 b.1 Electromagnetic Waves . . . 83
b.2 Interferometry . . . 85
b.3 Gaussian beams . . . 86
b.4 Lasers . . . 88
b.4.1 Stimulated emission and Einstein’s coefficient . . . 88
b.4.2 Population inversion . . . 90
c e l e m e n t s o f q ua n t u m o p t i c s 93 c.1 Quantization of the electromagnetic field . . . 94
c.2 Information encoding using photons . . . 95
Bibliography 99
L I S T O F F I G U R E S
Figure 2.1 Bloch sphere representation of a single qubit . . . 6
Figure 2.2 Sketch of a Bell experiment . . . 9
Figure 2.3 Sketch of the no-signaling (NS), quantum (Q), and local (L) sets 13 Figure 2.4 Schematic diagram of a generalized measurement . . . 17
Figure 3.1 Bell scenario involving a single Alice and multiple Bobs . . . 20
Figure 4.1 Schematic representation of the SPDC process . . . 24
Figure 4.2 SPDC energy and momentum constraints . . . 27
Figure 4.3 Experimental scheme of the source of polarization-entangled photons based on a polarization Sagnac interferometer . . . . 28
Figure 4.4 Implementation of the entangled photon surce . . . 29
Figure 5.1 Example of a trivial quantum circuit . . . 32
Figure 5.2 Scheme of the physical system analysed in our experiment . . 33
Figure 5.3 Expected CHSH inequality expectation values for the analyzed physical system . . . 43
Figure 5.4 Experimental design D . . . 44
Figure 5.5 Scheme of the experimental setup S . . . 49
Figure 5.6 Picture of Alice’s measurement setup . . . 51
Figure 5.7 Picture of Bob’s measurement setup . . . 51
Figure 5.8 Picture of Charlie’s measurement setup . . . 52
Figure 6.1 Glass Slide characterization graph . . . 57
Figure 6.2 Half Wave Plate characterization graph . . . 58
Figure 6.3 Polarization shift characterization graph . . . 59
Figure 6.4 Graph of the phase shift variation VS HWP tilting angle . . . 61
Figure 6.5 ∗phase displacement plot . . . 62
Figure 6.6 stability graph . . . 62
Figure 6.7 Fit parameter variation over a thirteen hours time interval . . 63
Figure 6.8 IACCHSHand IABCHSHVS plot . . . 68
Figure 6.9 Sequential violation of double Bell inequalities . . . 69
Figure A.1 Simple representation of a qubit wire. . . 79
Figure A.2 An arbitrary single-qubit gate. . . 80
Figure A.3 Controlled-U operation . . . 81
Figure A.4 Measurement circuit block . . . 81
Figure B.1 Interference phasor diagram . . . 85
Figure B.2 Simple rapresentation of a gaussian beam properties . . . 88
Figure B.3 Mechanisms of atomic transition . . . 89
Figure B.4 Population inversion . . . 90
Figure B.5 Sketch of a laser resonant optical cavity . . . 91
Figure C.1 Sketch of an electromagnetic wave . . . 96
Figure C.2 Wave Plate used as polarization basis rotator . . . 97
Figure C.3 Wave plate used as phase shifter . . . 98
Figure C.4 Quantum mechanical description of the beam-splitter . . . 99
1
I N T R O D U C T I O N
At the end of the 19thcentury physics, intended as the scientific method applied to natural sciences, had achieved remarkable results in the comprehension and explanation of the reality (classical mechanics, classical electromagnetism, classical thermodynamics). At that time the common feeling in the scientific community was a general satisfaction and the belief that almost every aspect of the physical world had already been discovered, and that the explanation of the phenomena still to understand would have been given by a finest and more accurate study of the physical laws already discovered.
The scientific community thought that the still open issues of the theory developed back then were just limited "problems", whose solution wouldn’t have affected much of the physical theoretic framework already developed. Two open issues seeking for an explanation were the black-body radiation and the photoelectric effect. The first one was pointed out by Rayleigh–Jeans, whom realised that classical statistical mechanics and classical electromagnetism predicted that a black body in thermal equilibrium would have emitted a radiation with infinite amount of energy, a clearly non-physical result. The second one was an experimental fact first noticed by H.
Hertz in 1887 whom wasn’t able to explain the frequency-threshold-effect observed in the emission of charged particles by a metal illuminated by an electromagnetic wave with the classical electromagnetic theory.
The two problems were solved independently at the beginning of the 20thcentury respectively by M. Plank (who was able to correctly predict the black-body radiation spectra introducing a quantization in the interaction between the black-body and the EM field) and by A. Eninstein (who was able to explain the photoelectric effect introducing a quantization of the EM field and assuming the atomic nature of matter).
These two discoveries opened a completely new research field for physicist, and revealed to the scientific community that the theoretical tools developed until that moment to understand reality were completely inadequate to explain a vast class of phenomena that involved small-scale systems. Since these two pioneering discoveries, some of the greatest physicist of that time (N. Bohr, L. de Broglie, A. H. Compton, P.
A. M. Dirac, E. Schrödinger, M. Born, W. Pauli, W. Heisenberg to name a few) worked on the development of new tools to explain physical reality, placing the building blocks of a new physical theory, Quantum Mechanics, that would change forever our conception of the world and our idea of how the mechanisms that rule nature work. The initial development of the theory concluded in 1930 with the publication of “The principles of Quantum Mechanics” by P. A. M. Dirac [8], a book where all the new discoveries about Quantum Mechanics were fixed and formalized. Quantum Mechanics, with all its successive developments, had an enormous impact in physics, and lays at the basis of all the most modern and sophisticated physical theories
i n t r o d u c t i o n
(Quantum Field Theory (QFT), Quantum Electrodynamic (QED) and the Standard Model).
One of the most recent developments of Quantum Mechanics has been its ap- plication to Information Theory. Information theory is the subject that studies the quantification, storage, and communication of information; its foundation dates back to 1948, with the publication of the landmark paper by C. E. Shannon “A Mathematical Theory of Communication” [9]. Its application to the development of new technologies is deeply affecting our everyday life, and it is the biggest responsible for all those innovations that are shifting us towards a new model of “communication society”.
Quantum Information is the study of the information processing tasks that can be accomplished using quantum mechanical systems. The use of quantum systems as information carriers introduces a completely new paradigm for information theory, leading to the development of new protocols for both computing and cryptography.
It is worth noting that, since its initial development in the 80s, quantum information has given a great quantity of remarkable theoretical results, whose experimental implementation is still at its infancy, due to the difficulties in the implementation an control of quantum systems. Nevertheless, in the recent years, a big technological im- provement has pushed for unprecedented experimental results [10] [11] [12] [13] [14] [15] [16] that make it reasonable to believe that a future where quantum technology will have an impact on every day life is not that far away.
The aim of this thesis fits perfectly with the scenario just described, since we wish to investigate experimentally a fundamental characteristic of quantum systems, some- thing that is sometimes considered the “the characteristic trait” [17] that distinguishes quantum mechanics from the classical theory: non-locality. In the following sections we will better define what non-locality is, for the moment we just point out that our work is one of first experimental implementations of a device capable of detecting non-local correlations between three observers sharing a two-photon entangled state.
The purpose of this work is double: in first place we want to shed some new light on non-local correlations, demonstrating a property until now just theoretically pre- dicted. This is in fact one of the first works to obtain experimentally this result. Only at the final stage of this project a study with similar results has been published [18].
THe second intent of this thesis is to build a stable apparatus capable of measuring multiple consecutive non-local correlations; this result may have important applica- tions for Quantum Random Number Generation or for Quantum Key Distribution.
In the following we will describe briefly the structure of this thesis.
In the first chapter we will focus on the concept of non-locality. Firstly we will introduce the qubit, the basic unit of quantum information. This will give us the opportunity to introduce some aspects of quantum mechanics of interest for the rest of the thesis. We will then explain the concepts of non-locality, entanglement and weak measurements.
In the second chapter we will focus on the theoretical framework of our work. We will discuss Bell inequalities and CHSH correlations, the most common tools used to prove the presence of non-local correlations between a multiple system. We will then present and characterize a scenario where non-local correlations are shared among a number of observers larger then two. We will finally discuss the possible applications of such multiple non-local correlations.
i n t r o d u c t i o n
In the third chapter we will describe how a photonic entangled state can be experimentally produced, studying the physics of the process. We will then describe the setup actually employed in our experiment.
In the fourth chapter we will describe theoretically and practically our experiment.
In the fifth chapter we will show the obtained results, reporting both the characteri- zation of theapparatus and the results of our experiment.
In the sixth chapter the conclusion of our work will be given.
Finally, in the appendixes, we will briefly review the fundamental tools needed to understand this work, discussing about: Quantum Mechanics, Classical Optics and Quantum Information.
2
N O N - L O C A L I T Y A N D W E A K M E A S U R E M E N T
In this chapter we will discuss some of the most fundamental aspects Quantum Mechanics we will be dealing with in our work. We will start introducing the qubit, the basic unit of quantum information. A deep comprehension of its properties is fundamental because we will often use a qubit to make examples in our our brief quantum mechanics explanation and because it is the physical state experimentally involved in our work. We will then introduce qubits composite systems and we will present the concepts of entangled states, non-locality and Weak Measurements.
In the following sections we will intensively make use of the concepts and tools presented in appendixA on page 75.
2.1 q u b i t
The basic unit of classical information theory is the bit, an object that can assume two values, 0 or 1. Similarly, quantum information theory has adopted its quantum counterpart, called qubit (quantum bit) that is a two-level system described within the framework of quantum mechanics. Because of Quantum Mechanics Postulate1, such a system corresponds to a two-dimensional Hilbert spaceH ≈ C2, with basis vectors
|0i and |1i. The great difference between bits and qubits is that a qubit can be in a state other than|0i or |1i; indeed, any linear combination of these two states (called superposition) is allowed. The general qubit state|ψi therefore is written as:
|ψi = α |0i + β |1i (2.1)
where α and β are complex numbers such that|α|2+|β|2 = 1. Postulate1states also that quantum states are defined up to a global phase factor (the vectors|ψi and eiγ|ψi describe the same physical state). Therefore, it is possible to take as representative of the physical state the vector with α ∈R. This, together with the requirement of normalization, allows us to write the state of a single qubit as
|ψi = cosθ
2|0i + eiφsinθ 2 |1i
with θ and φ real numbers. These numbers define a point on the surface of the unit three-dimensional sphere, the Bloch sphere, shown in figure2.1. In this representation, the qubit |ψi is associated with the point (sin θ cos φ, sin θ sin φ, cos θ). The Z axis corresponds to the computational basis{|0i , |1i}, while the two other axes are associated to the diagonal basis X ≡
|+i = |0i+|1i√2 ,|−i = |0i−|1i√2
and the circular basis Y ≡ |ri = |0i+i|1i√2 ,|li = |0i−i|1i√2
[4].
n o n-locality and weak measurement
Y X
Z
Figure 2.1.: Bloch sphere representation of a single qubit.
Another useful qubit representation is the matrix one, that associates the vectors of the computational basis with
|0i =1 0
|1i =0 1
In equation2.1the qubit is in a pure state. A more general representation of the qubit can be given using the density matrix formalism; with this formalism a qubit (both in a pure or in a mixed state) can be written as:
ˆρ = 12+r · σ 2
where r is a real three-dimensional vector such that||r|| 6 1 and σ is the “vector” of the three Pauli matrices
ˆσ1= ˆσx=
0 1 1 0
ˆσ2 = ˆσy=
0 −i i 0
ˆσ3 = ˆσz=
1 0 0 −1
as components. The vector r corresponds a point in the Bloch sphere. It can be shown that a pure state is characterized by a vector r on the surface of the sphere (||r|| = 1), while a true mixed state is represented by a point inside the sphere.
It is possible to use any two-level quantum system in order to create a physical qubit. For example, the spin of an electron or two electronic levels of an atom are suitable for qubit realization. For our purpose, the most important physical system to realize a quantum bit is the photon, the the electromagnetic field quantum.
2.2 entangled states and epr paradox
2.1.1 Multiple qubit systems
The difference between classical and quantum information is more marked when dealing with compound systems. Classically, the composition of n systems is de- scribed by an n-bit string of 0s and 1s (e.g. the composition of 8 bits is described by an 8-bit string called byte). In the quantum case, on the other hand, things are slightly more complicated. Quantum mechanics Postulate2says that the compound state of two systems lies in the tensor product of the two Hilbert spaces describing the single systems. Therefore, if the i-th qubit lies inHi≈C2 space, the state describing the composition of n qubits is described by a vector in:
H = H1⊗ · · · ⊗Hn≈C|2⊗ · · · ⊗{z C2}
ntimes
≈C2n.
Like single-qubit systems, multiple-qubit systems can be represented using 2n- component complex vectors.
2.2 e n ta n g l e d s tat e s a n d e p r pa r a d o x
Postulate2enables us to introduce one of the most interesting and puzzling ideas associated with composite quantum systems and quantum mechanics in general:
entanglement. If we consider the two qubit state
|ψi = |00i + |11i√ 2
we can easily notice that there are no single qubit states |ai and |bi such that
|ψi = |ai |bi. This example leads us to the definition of the simple and hard-to- understand concept of entanglement: a state of a composite system that can’t be written as the product of states of its component systems is defined as an entangled state.
The properties of an entangled system are among the most challenging conse- quences of quantum mechanics. It was studying them that, for example, Einstein, Podolski and Rosen (ofter reffered as EPR) in their famous article published in 1935
“Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?” [1] arose what they considered a leak in quantum mechanics theory. In their paper the authors criticised the Copenhagen interpretation of quantum mechanics, accusing quantum mechanics to be an incorrect (and at least incomplete) tool to describe physical reality. Their argument, as reformulated by Bhom [19] some years later, can be summarized as follows: let’s consider a spin-zero particle decaying into two spin-half particles such that there is no interaction between them after decay. The quantum state of the two particles before to measurement can be written as:
|ψABi = |↑iA|↓iB√−|↓iA|↑iB 2
Here, subscripts A and B distinguish the two particles, which can be thought to be in the possession of two experimentalists called Alice and Bob1. The rules of quantum
1 The reformulation of quantum physics problems in terms of measurements on the system by some observers is a convention generally adopted in Quantum Information.
n o n-locality and weak measurement
theory predict the outcomes of the measurements performed by the experimentalists.
Alice, for example, will measure her particle to be spin-up in half of her measurements, Alice’s measurement causes the state of the two particles to collapse, so that if Alice measures spin-up in some direction n, the quantum state after the measurement is the corresponding eigenstate:
|ψABi =|↑nnniA|↓nnniB
If Bob also measures spin in direction n, he must get a spin-down result. Hence, spin measurements in the same direction are always anti-correlated, even if the particles are spatially separated, meaning that no signal can be exchanged among them. EPR saw this as evidence of the incompleteness of quantum theory: if there is no interaction between particles, then the only explanation for this anti-correlation between measurement outcomes is that each particle carries a pre-existing determinate value (appropriately anti-correlated with the value carried by the other particle) for that measurement. Such a property is unaccounted for by the quantum mechanical state description, and their paper concludes:
While we have thus shown that the wave function does not provide a complete description of the physical reality, we left open the question of whether or not such a description exists. We believe, however, that such a theory is possible.
The goal of EPR was to show that quantum mechanics is incomplete, by demonstrating that quantum mechanics lacked some essential ’element of reality’, according to their criterion. They hoped to return to a more classical view of the world, in which systems have properties which exist independently of the measurements performed on them.
Since the EPR paper the scientific community got really interested in the argument, and divided in two groups: one (those who agreed with EPR thesis) which tried to solve the paradox working on a new kind of theories (hidden variables theories) that could reconcile quantum mechanics results with a more classical vision of reality, the other one which worked on finding an acceptable interpretation of quantum mechanics that could justify its results to our "classical" eyes.
A fundamental advancement in this topic was given by the article “On the Einstein- Podolsky-Rosen Paradox” by J. S. Bell published in 1964 [3] which demonstrated that predictions of quantum theory are incompatible with those of any physical theory satisfying a natural notion of locality. Bell’s argument will be exposed in the following sections.
2.3 n o n-locality, bell inequalities and chsh inequality
In his 1964 article, Bell proved that there is no Local Hidden Variable theory that can reproduce Quantum Mechanics results. Bell’s theorem has deeply influenced our perception and understanding of physics, and arguably ranks among the most profound scientific discoveries ever made. In the following we will present the Bell theorem considering some properties of a thought experiment; we will take advantage of its explanation to introduce the concept of non-locality. The explanation closely follows the one given by Brunner, Cavalcanti, Pironio et al. in the review "Bell nonlocality" [20].
2.3 non-locality, bell inequalities and chsh inequality
Figure 2.2.: Sketch of a Bell experiment. A source (S) distributes two physical systems to distant observers, Alice and Bob. Upon receiving their systems, each observer performs a measurement on it. The measurement chosen by Alice is labeled x and its outcome a. Similarly, Bob chooses measurement y and gets out- come b. The experiment is characterized by the joint probability distribution P(ab|xy) of obtaining outcomes a and b when Alice and Bob choose measurements x and y. [20]
In a typical Bell experiment, two systems which may have previously interacted are spatially separated and are measured by two distant observers, Alice and Bob (see figure2.2). Alice may choose to perform one of several possible measurements on her system. Let x denote her measurement choice. For instance, x may refer to the position of a knob on her measurement apparatus. Similarly, let y denote Bob’s measurement choice. Once the measurements are performed, they give outcomes a and b on the two systems. The actual values assigned to the measurement choices x, y and outcomes a, b are purely conventional; they are mere macroscopic labels distinguishing the different possibilities.
For each run of the experiment, the outcomes a and b may vary, even when the same choices of measurements x and y are made. These outcomes are thus in general governed by a probability distribution P(ab|xy), which can of course depend on the particular experiment being performed. By repeating the experiment a sufficient number of times and collecting the observed data, one gets a fair estimate of such probabilities. When such an experiment is actually performed (for example by generating pairs of spin-1/2 particles and measuring the spin of each particle in different directions) it will in general be found that
P(ab|xy) 6= P(a|x)P(b|y)
implying that the outcomes on both sides are not statistically independent from each other. Even though the two systems may be separated by a large distance, and may even be spacelike separated, the existence of such correlations is nothing mysterious.
In particular, it does not necessarily imply some kind of direct influence of one system on the other, for these correlations may simply reveal some dependence relation between the two systems which was established when they interacted in the past.
This is at least what one would expect in a local theory. Being more precise, the assumption of locality implies that we should be able to identify a set of past factors, described by some variables λ, having a joint causal influence on both outcomes, and which fully account for the dependence between a and b. Once all such factors have been taken into account, the residual indeterminacies about the outcomes must now be decoupled; that is, the probabilities for a and b should factorize:
P(ab|xy, λ) = P(a|x, λ)P(b|y, λ)
n o n-locality and weak measurement
This factorability condition simply expresses the fact that we have found an explana- tion according to which the probability for a depends only on the past variables λ and on the local measurement x, but not on the distant measurement and outcome, and analogously for the probability to obtain b. The variable λ will not necessarily be constant for all runs of the experiment, even if the procedure which prepares the particles to be measured is held fixed, because λ may involve physical quantities that are not fully controllable. The different values of λ across the runs should thus be characterized by a probability distribution q(λ). Combined with the above factorability condition, we can thus write:
P(ab|xy) = Z
Λ
q(λ)P(a|x, λ)P(b|y, λ)dλ (2.2) where we also implicitly assumed that the measurements x and y can be freely chosen in a way that is independent of λ, i.e., that q(λ|x, y) = q(λ). This decomposition now represents a precise condition for locality in the context of Bell experiments. Note that no assumptions of determinism or of a “classical behavior” are being involved in equation2.2: we assumed that a (and similarly b) is only probabilistically determined by the measurement x and the variable λ, with no restrictions on the physical laws governing this causal relation. Locality is the crucial assumption behind equation2.2.
In relativistic terms, it is the requirement that events in one region of space-time should not influence events in spacelike separated regions.
It is now straightforward to prove that the predictions of quantum theory for certain experiments involving entangled particles do not admit a decomposition of the form 2.2. To establish this result, we consider for simplicity an experiment where there are only two measurement choices per observer x, y ∈{0, 1} and where the possible outcomes take also two values labeled a, b ∈{−1, +1}. Let haxbyi =P
a,babP(ab|xy) be the expectation value of the product ab for a given measurement choice (x; y) and consider the expression S = ha0b0i + ha0b1i + ha1b0i − ha1b1i, which is a function of the probabilities P(ab|xy). If these probabilities satisfy the locality decomposition (2.2), we necessarily have that
S =ha0b0i + ha0b1i + ha1b0i − ha1b1i 6 2 (2.3) which is known as the Clauser-Horne-Shimony-Holt (CHSH) inequality [21].
To derive this inequality, we can use equation2.2in the definition of haxbyi, which allows us to express this expectation value as an average haxbyi =R
Λq(λ)haxiλhbyiλdλ of a product of local expectations haxiλ =P
aaP(a|x, λ) and hbyiλ =P
bbP(b|y, λ) taking values in [−1, +1]. Inserting these expressions into equation2.3, we can write:
S = Z
Λ
q(λ)Sλdλ with
Sλ=ha0iλhb0iλ+ha0iλhb1iλ+ha1iλhb0iλ−ha1iλhb1iλ Since ha0iλ, ha1iλ∈ [−1, +1], this last expression is smaller than
Sλ6 |hb0iλ+hb1iλ| + |hb0iλ−hb1iλ|
Without any loss of generality we can assume that hb0iλ > hb1iλ > 0 which yelds Sλ= 2hb0iλ6 2, and thus S =R
Λq(λ)Sλdλ6 2.
2.3 non-locality, bell inequalities and chsh inequality
If we consider now the quantum predictions for an experiment in which the two systems measured by Alice and Bob are two qubits in the singlet state|Ψ−i = |01i−|10i√
2 ,
where|0i and |1i are the eigenstates of σzfor the eigenvalues +1 and −1, respectively.
Let the measurement choices x and y be associated with vectors xxxand yyycorresponding to the measurement of xxx· σσσon the first qubit and of yyy· σσσon the second qubit.
According to quantum theory, we then have the expectations haxbyi = −xxx· yyy. If we choose the two measurement settings x ∈{0, 1} correspond to measurements in the orthogonal directions eee1 and eee2 respectively, and the two measurement settings y∈{0, 1} correspond to measurements in the diagonal directions −eee1√+e2ee2 and eee1√−e2ee2, we get that:
ha0b0i = ha0b1i = ha1b0i = 1
√ 2 ha1b1i = − 1
√ 2 obtaining finally that:
S = 2√ 2 > 2
in contradiction with equation2.3and thus with the locality constraint (2.2). This is the content of Bell’s theorem, establishing the non-local character of quantum theory and of any model reproducing its predictions. The CHSH inequality (2.3) is an example of a Bell inequality, a linear inequality for the probabilities P(ab|xy) that is necessarily verified by any model satisfying the locality condition (2.2), but which can be violated by suitable measurements on a pair of quantum particles in an entangled state. The violation of these inequalities and the predictions of quantum theory were first confirmed experimentally by Freedman and Clauser [22], then more convincingly by Aspect, Grangier, and Roger [23], and in many other experiments since then.
2.3.1 Mathematical characterization of non-local correlations
In the previous section we introduced the concept of non-locality, describing it as a characteristic property of entangled quantum system that emerges from the outcomes correlations. In this section we want to formalize the concept, showing how is possible to characterize the argument mathematically. We suggest to refer to [20] for an exhaustive treatment of the argument.
We start considering a situation similar to the one presented in the previous section.
We consider two distant observers, Alice and Bob, performing measurements on a shared physical system, for instance, a pair of entangled particles. Each observer has a choice of m different measurements to perform on his system. Each measurement can yield ∆ possible outcomes. Abstractly we describe the situation by saying that Alice and Bob have access to a "black box". Each party locally selects an input (a measurement setting) and the box produces an output (a measurement outcome). We refer to this scenario as a Bell scenario.
We label the inputs of Alice and Bob x, y ∈ {1, . . . , m} and their outputs a, b ∈ {1, . . . , ∆} respectively. The labels attributed to the inputs and outputs are purely conventional, and the results presented here are independent of this choice.
n o n-locality and weak measurement
Let P(ab|xy) denote the joint probability to obtain the output pair (a, b) given the input pair (x, y). A Bell scenario is then completely characterized by ∆2m2 such joint probabilities, one for each possible pair of inputs and outputs. We refer to the set PPP ={P(ab|xy)} of all these probabilities as a behavior. Informally, we simply refer to them as the correlations characterizing the state shared by Alice and Bob. A behavior can be viewed as a point PPP∈R∆2m2 belonging to the probability spaceP ⊂ R∆2m2 defined by the positivity constraints P(ab|xy) > 0 and the normalization constraints P∆
a,b=1P(ab|xy) = 1. Due to the normalization constraints P is a subspace of R∆2m2 of dimension dimP = (∆2− 1)m2.
The existence of a given physical model behind the correlations obtained in a Bell scenario translates into additional constraints on the behaviors PPP. Three main types of correlations can be distinguished:
1. No-signaling correlations;
2. Local correlations;
3. Quantum correlations.
No-signaling correlations
The first natural limitation on behaviors PPP are the no-signaling constraints, formally expressed as:
X∆ b=1
P(ab|xy) = X∆ b=1
P(ab|xy0) for all a, x, y, y0 X∆
a=1
P(ab|xy) = X∆ a=1
P(ab|x0y) for all b, y, x, x0
These constraints have a clear physical interpretation: they imply that the local marginal probabilities of Alice P(a|x) ≡ P(a|xy) =P∆
b=1P(ab|xy) are independent of Bob’s measurement setting y, and thus Bob cannot signal to Alice his choice of input (and the other way around). In particular, if Alice and Bob are spacelike separated, the no-signaling constraints guarantee that Alice and Bob cannot use their black box for instantaneous signaling, preventing a direct conflict with relativity.
We denote withNS the set of behaviors satisfying the no-signaling constraints.
Local correlations
A more restrictive constraint than the no-signaling condition is the locality condition.
Formally, the set L of local behaviors is defined by the elements of P that can be written in the form (as already introduced in equation2.2):
P(ab|xy) = Z
Λ
q(λ)P(a|x, λ)P(b|y, λ)dλ
where the (hidden) variables λ are arbitrary variables taking value in a space Λ and distributed according to the probability density q(λ) and where P(a|x, λ) and P(b|y, λ) are local probability response functions for Alice and Bob, respectively.
Operationally, one can also think about λ as shared randomness; that is, some shared classical random bits, where Alice will choose an outcome a depending on both her measurement setting x and λ and similarly for Bob.
2.3 non-locality, bell inequalities and chsh inequality
Whereas any local behavior satisfies the no-signaling constraint, the converse does not hold. There exist no-signaling behaviors which do not satisfy the locality conditions. Hence the set of local correlations is strictly smaller than the set of no-signaling correlations; that is,L ⊂ NS.
Quantum correlations
Finally, we consider the set of behaviors achievable in quantum mechanics. Formally, the setQ of quantum behaviors corresponds to the elements of P that can be written as:
P(ab|xy) = Tr(ρABMa|x⊗ Mb|y) (2.4) where ρABis a quantum state in a joint Hilbert spaceHA⊗HBof arbitrary dimension, Ma|x are measurement operators [positive operator valued measure (POVM) ele- ments] onHAcharacterizing Alice’s measurements (thus Ma|x > 0 andP∆
a=1Ma|x = 1), and similarly Mb|y are operators onHB characterizing Bob’s measurements.
It can easily be shown that any local behavior admits a description in terms of equation2.4and thus belongs toQ (L ⊂ Q). Moreover, any quantum behavior satisfies the no-signaling constraints (Q ⊂ NS). However, there are quantum correlations that do not belong to the local set (this follows from the violation of Bell inequalities, Q * L) and, there are no-signaling correlations that do not belong to the quantum set (NS * Q).
Bell inequalities
The setsL, Q and NS are closed, bounded, and convex. That is, if PPP1 and PPP2 belong to one of these sets, then the mixture µPPP1+ (1 − µ)PPP2 with 0 6 µ 6 1 also belongs to this set. By the hyper-plane separation theorem, it follows that for each behavior PPPˆ ∈ R∆2m2 that does not belong to one of the setsK = L, Q, or NS there exists a hyperplane that separates this ˆPPPfrom the corresponding set (see figure 2.3). That is,
Figure 2.3.: Sketch of the no-signaling (NS), quantum (Q), and local (L) sets. Notice the strict inclusions L ∈ Q ∈ NS. Moreover, NS and L are polytopes, i.e., they can be defined as the convex combination of a finite number of extremal points. The setQ is convex, but not a polytope. The hyperplanes delimiting the set L correspond to Bell inequalities. [20]
n o n-locality and weak measurement
if ˆPPP /∈K, then there exists an inequality of the form sss· PPP = X
a,b,x,y
sabxyP(ab|xy) 6 Sk
that is satisfied by all PPP∈K but which is violated by ˆPPP : sss · ˆPPP > Sk.
In the case of the local setL, such inequalities are simply Bell inequalities. Thus any non-local behavior violates a Bell inequality. An example of such an inequality is the CHSH inequality (see equation2.3) introduced in section2.3. The inequalities associated with the quantum set, which characterize the limits ofQ, are often called quantum Bell inequalities or Tsirelson inequalities.
We conclude here the introduction to the mathematical characterization of the possible physical correlations in a two observer scenario experiment. To the scope of this thesis is sufficient to have clear what we mean with the correlations setsL, Q and NS, and what we mean with Classical bound and Tsirelson’s bound. For further studies and a more in depth understanding of the subject we refer to [20], where a very good introductory explanation and a comprehensive bibliography of the argument can be found.
2.4 w e a k m e a s u r e m e n t
The last concept I want to introduce before discussing the main argument of this work is the concept of weak measurement. The first fact to point out is that there is no universally accepted definition of a weak measurement even though the idea at the basis of this concept is very simple: weak measurements are a type of quantum measurement where an observer obtains very little information about the system, but also disturbs the state very little. We know that any measurement on a system necessarily disturbs it (as formalized in the Busch’s theorem [24]); with weak measurements we gain some informations about the system, without compromising (and then defining) completely its state. In this picture it is clear that:
• the more a measurement is weak, the less information we obtain about the system, and the less we perturb it;
• the less a measurement is weak, the more it is strong, and the more we approach a projective measurement, where we obtain all the possible information about the state (getting the outcome of the measurement) and where we drastically perturb it (projecting it onto the eigenspace of the measurement operator) [25].
The key aspect of the topic is the trade-off between the amount of information we can get about a quantum system and the amount of disturbance we introduce on it.
Since the birth of quantum mechanics this has been a largely studied topic, and a lot of literature has been produced exploring the subject, even if the definition "weak measurement" is quite recent and was probably used for the first time in 1988 by Aharonov et al. [26].
The aim of this thesis is not that of reviewing exhaustively the subject, we are rather interested in giving an idea of what we mean by weak measurements, and how we can perform them, since our goal is to exploit them to correlate weakly two systems enabling for a double CHSH inequality violation. More information, and a deeper review of the argument, can be found in [27, 28,29,30]. In our exposition we will
2.4 weak measurement
closely follow the approach of B. E. Y. Svensson in his review [31].
Our explanation of weak measurements starts with a generalization of the quantum strong measurement (presented in5on page 76), that takes into account not only the quantum state under study, but also the practical implementation of the measurement, by introducing in our picture the measurement device (from now on called ancilla or meter) that couples with the studied system to allow the extraction of information. This ancilla-scheme goes a few steps in the direction of describing the very measurement process, highlighting some aspects of measurement that are classically left uncovered such as: what kind of measurement apparatus is used and what distinguishes measurements from other possible types of interactions.
It is the interaction of the ancilla with the system (called also the object) that constitutes the measurement: by reading off the meter one gets information about the value of the system observable. In some experimental situations, the meter can even be a property of the object under analysis different from than the one (system) we are interested into (like momentum and polarization for a photon).
2.4.1 Modelling the ancilla measurement process
The meterM will be modeled as a quantum device living in a Hilbert space HM having a complete, orthonormal set of basis states |mki, k = 1, 2, . . . , dM that are eigenvectors of the operator ˆM. The intrinsic Hamiltonian ofM is denoted ˆHM. The meter is assumed to be prepared in an initial pure state |m(0)i, so that the meter initial density matrix is ˆµ0 =|m(0)i hm(0)|.
The object or system S has its Hilbert space HS, and has its complete, orthonormal set of basis states |sii, s = 1, 2, . . . , dS. Such set of vectors are eigenstates of the operator ˆS inHSwhich corresponds to the observable S to be measured. The intrinsic Hamiltonian of the system is ˆHM. The system is assumed to be initially prepared (pre-selected) either in a pure state|si (in which case its density matrix is ˆσ0=|si hs|) or in a more general state described by an arbitrary ˆσ0.
The total system T comprises the object-system S and the meter M. Its Hilbert space isHT=HS⊗HM and the initial state of the total system is ˆτ0= ˆσ0⊗ ˆµ0, the system and the meter are assumed to be initially uncorrelated (not entangled). The total Hamiltonian is
HˆT= ˆHS+ ˆHM+ ˆHint
and we consider that the only non vanishing term in ˆHT is ˆHint, the interaction Hamiltonian betweenS and M.
2.4.2 The pre-measurement
The system and the meter are assumed to interact via a unitary time-evolution operator ˆUin what is called a pre-measurement. This means that the total systemT with its initial density matrix ˆτ0 will evolve unitarily into ˆτ1:
ˆτ0 = ˆσ0⊗ ˆµ0
Uˆ
−→ ˆτ1 = ˆUˆσ0⊗ ˆµ0Uˆ†
n o n-locality and weak measurement
where ˆU† is the Hermitian conjugate of ˆU. For completeness we point out that the unitary time evolution operator ˆUis linked to the Hamiltonian by:
U = eˆ − hi Rdt ˆHT
for ˆU to be a pre-measurement of ˆS, ˆU must be able to distinguishes between the different states|sii. It is therefore assumed that an initial joint pure state|sii ⊗|m(0)i of the system and the meter is transformed by ˆUinto:
|sii ⊗|m(0)i −U→ˆ Uˆ|sii ⊗|m(0)i
=|sii ⊗|m(i)i
where i = 1, 2, . . . , dS and the meter states|m(i)i act as markers for the system state
|sii; we will see in detail how this comes about later.
If this initial state is a superposition of eigenstates, the linearity of ˆUgives:
|si ⊗ |m(0)i −U→ˆ Uˆ|si ⊗ |m(0)i
=
dS
X
i=1
ci|sii ⊗|m(i)i (2.5) or, in the density matrix notation:
ˆτ0= ˆσ0⊗ ˆµ0 Uˆ
−→ ˆτ1 = ˆUˆσ0⊗ ˆµ0Uˆ†
=X
i,j
hsi| ⊗ hm(i)| hsi| ˆσ0|sji hsj| ⊗ hm(j)|
=X
i,j
hm(i)| ˆPsiˆσ0Pˆsjhm(j)| with ˆPsi projectors in theHS space.
We note that:
• A system’s pure eigenstate|sii is left unchanged under this operation.
• One of the most important consequences of the pre-measurement is that the system’s state becomes correlated (entangled) with the meter state: ˆτ1 cannot be written as a product of one object state and one meter state (2.5).
• The meter states|m(0)i and|m(i)i are, in general, not eigenstates of the meter operator ˆM, but superpositions of such eigenstates. In particular, the states
|m(0)i and|m(i)i, i = 1, 2, . . . , dS, are normalized but in general not mutually orthogonal. Nor do they form a complete set in HM. Indeed, the dimensions dS and dM of the respective Hilbert spacesHS andHMneed not be equal.
• The operation ˆUthus correlates the system state|sii with the meter state|m(i)i but not necessarily in a unique way: to each |sii there corresponds a definite
|m(i)i, different for different|sii, but there could be overlap between|m(i)i and
|m(j)i, expressed by hm(i)|m(j)i 6= 0, for i 6= j.
The rule for obtaining the separate states ˆσ1 for the system, and ˆµ1 for the meter, after this pre-measurement, is to take the partial trace over the non-interesting degrees of freedom (A.2.1). In case we want the state ˆσ1 of the system, this means execute the partial trace over ˆHM:
ˆσ0 Uˆ
−→ ˆσ1= T rMˆτ1 =X
k
hmk| ˆτ1|mki
=X
i,j
ˆPsiˆσ0Pˆsjhm(j)|m(i)i
2.4 weak measurement
Figure 2.4.: Schematic diagram of a generalized measurement. The system of interest is coupled to an ancilla pre- pared in a known state ˆµ0by the unitary evolution ˆU, and then a projective measurement is performed on the ancilla.
As is seen, in case of no overlap between different states|m(i)i, the ancilla-measurement scheme have the same consequences for the object-system as would a projective mea- surements scheme would have had. The general case, hm(i)|m(j)i 6= 0 for i 6= j, does allow for interference between different eigenstates associated to the eigenvalues si, a fact which will have interesting measurable effects.
2.4.3 The read-out
So far, no real measurement has been performed in the sense of obtaining a record.
The entangled system-meter is still in a quantum-mechanical superposition ˆτ1. One needs a recording, a read-out of the meter, in order to obtain information that constitutes a real measurement. Therefore, the next step in this ancilla measurement scheme is to subject the meter, and only the meter, to a projective measurement of the pointer observable M (see figure 2.4 for a schematic representation of the measurement process). Reading off the meter means obtaining an eigenvalue mk
of the corresponding operator ˆMin a operation that is symbolized by the projector Oˆmk =|mki hmk| onto the corresponding subspace of the meter Hilbert space HM. Since, as a result of the pre-measurement, the system is entangled with the meter, this will influence the system state too, and is therefore also a measurement of the object-system as will be evident shortly.
For the total density matrix, this projective measurement, implies:
ˆτ1
Oˆmk
−−−→ ˆτ1(|mk) = ˆIS⊗ ˆOmk
ˆτ1 ˆIS⊗ ˆOmk
P(mk)
where with ˆτ1(|mk) we mean the total density matrix state ˆτ conditioned by the reading of the outcome mk, ˆIS is the unit operator in HS and where P(mk) = P(mk| ˆτ1) is the probability of obtaining the ancilla value mk given the state ˆτ1.
The good question to ask now is if we have learned something concerning the system state from the read-out of the meter. To answer this question we have to look at the density matrix of the system ˆσ1(|mk) after read-out, that tells what is the
n o n-locality and weak measurement
system status after that action. Such density matrix matrix is obtained by partially tracing the overall system density matrix:
ˆσ1(|mk) = T rM ˆτ1(|mk) (2.6)
= 1
P(mk)T rM
ˆIS⊗ ˆOmk
ˆτ1 ˆIS⊗ ˆOmk
(2.7)
= 1
P(mk)hmk| ˆU ˆσ0⊗ ˆµ0Uˆ†|mki (2.8)
= 1
P(mk)hmk| ˆU |m(0)i ˆσ0hm(0)| ˆU†|mki (2.9)
= 1
P(mk)Ωˆkˆσ0Ωˆ†k (2.10)
where ˆΩk is an operator inHSdefined by:
Ωˆk=hmk| ˆU |m(0)i
=X
i
hmk|m(i)i|sii hsi|
=X
i
hmk|m(i)i ˆPsi
The operators ˆΩk are called measurement operators and are often denoted ˆMk in literature (here we use ˆΩk since in our notation M entitles entities related to the meter).
At this point, the ancilla-scheme approach to the measurement theory allows us to easily understand the novel interesting properties of what we called weak measurement. A quantum mechanical measurement, in general implies that the object-system under study will suffer large changes (disturbances) in its state, even if the measurement is considered non-destructive; in our formalism, these disturbances appear in the change from the initial density matrix ˆσ0 to a usually quite different matrix ˆσ1 after the measurement (see2.6). A weak measurement is a measurement that disturbs the state of the object of interest as little as possible. As we will see, a weak measurement is also such that the measurement results are less clear than in a strong or projective measurement. For example, there will be difficulties in distinguishing one eigenvalue of the observable under study from another. An interesting fact is that it has been shown that in certain conditions weak measurements allows for the emerging of new phenomena that can only be studied by weakening the interaction responsible for the measurement as much as possible [31].
In our work the result obtained in equation2.5is particularly meaningful; we saw that after the pre-measurement the two objects (system and ancilla) are entangled, and that this entanglement can be weakened as desired by appropriately choosing the m(i) basis. This result is the core idea of our work; our goal is in fact to weakly entangle a system and the ancilla in order to allow to share non-local correlations with a third system.
3
N O N - L O C A L I T Y S H A R I N G A M O N G M U LT I P L E O B S E RV E R S
In this chapter we will discuss the topic of interest of this thesis. In the previ- ous chapter we introduced the concepts of non-locality and weak measurement;
we characterized non-locality mathematically, and we showed how this property basically qualifies a class of correlated measurements between two observers. All the discussions we made in section2.3had a two-observer scenario (Alice and Bob) as reference situation. This is not a casual aspect: generally scientific literature on the argument focus on characterizing non-locality properties in a two-observer (or at most multipartite) scenario. Only very recently the possibility of sharing non-locality among multiple observers has been taken into account.
The aim of this work is to explore experimentally the topic of non-locality sharing among multiple observers, building one of the first apparatuses capable of proving the possibility to establish multiple non-local correlation among three observers sharing the same spin-1/2 entangled state.
This work is inspired by the article “Multiple Observers Can Share the Nonlocality of Half of an Entangled Pair by Using Optimal Weak Measurements” by Silva, Ralph, et al.
[7], which explored theoretically the possibility to observe such non-Locality sharing.
In the following we will introduce the topic closely following their argument.
3.1 t r i pa r t i t e b e l l i n e q ua l i t y
In the previous chapter, talking about measurement, we already discussed about one of the fundamental traits of quantum mechanics: in order to probe the properties of a system one must perturb it. We distinguished among two different types of measures:
• "Strong" measurements which collapse the system into one of the eigenstates of the measured observable; this type of measurements give the maximum information about the system.
• "Weak" measurements that disturb the system infinitesimally, giving only a small amount of information about the state.
It is interesting to study the situation where the strength of the measurement can vary continuously from very weak to strong, analysing how the trade-off between the degree of disturbance on the system and the amount of information gained evolves with the measurement strength. We will explore this topic considering measurements on a pair of entangled spin-1/2 particles, focusing on a new fundamental question in non-locality: can the non-locality of an entangled pair of particles be distributed among multiple observers, considered a scenario where the observers act sequentially and independently of each other?
n o n-locality sharing among multiple observers
Figure 3.1.: Bell scenario involving a single Alice and multiple Bobs, where the dashed lines indicate a spin-1/2 particle being transmitted, and the solid lines the inputs and outputs. [7]
We consider the scenario where a single observer has access to one of the particles of an entangled pair, and a group of two of observers have access to the second particle. Each observer in the second group acts independently, so the first one performs a measurement on the particle before passing it to the second one. We address the question of whether the single observer with the first particle can see non-local correlations with both the two observers that have access to the second particle. This Bell-scenario is represented in figure3.1.
In our scheme we will call Alice the observer that has exclusive access to one-half of the entangled pair of spin-1/2 particles, and we will call respectively Bob1 and Bob2 the two observers that have access to the other half of the the entangled pair. The Bobs are independent; i.e., Bob2 is ignorant of the direction that Bob1 measures his spin in as well as the outcome of his measurement.
We investigate whether the statistics of the measurements of Bob1 and Bob2 can both be non-local with Alice by testing the conditional probabilities P(ab1|xy1)and P(ab2|xy2)against the CHSH inequality.
At first one may think it impossible to have simultaneous violations Alice-Bob1
and Alice-Bob2 because of the monogamy of entanglement and of nonlocality [32].
However, these results assume no-signaling between all parties, while in our scenario Bob1implicitly signals to Bob2his choice of measurement on the state before he passes it on. Hence, no monogamy argument holds, and one has to look more closely at the situation.
We assume the measurements are unbiased; i.e., both Bobs choose the inputs 0 and 1with equal probability. Clearly Bob1cannot perform a strong measurement, since he would destroy the entanglement, and prevent Bob1from being non-local with Alice.
However, Bob1may not be able to observe non-locality with a very weak measurement either. To see this precisely, consider that Alice and the Bobs initially share a singlet state, and that they perform the standard measurements that attain Tsirelson’s bound for the CHSH inequality: i.e., Alice measures in the ¯Zor ¯Xdirection, corresponding to inputs 0 or 1, respectively, and the Bobs measure in the directions −Z+ ¯¯√2X or − ¯Z+ ¯√2X, for their respective inputs 0 or 1.
The Bobs receive a spin-1/2 particle, whose spin in the|Hi , |Vi polarization basis can be written:
|ψi = α |Hi + β |Vi
3.1 tripartite bell inequality
Bob1 is the first one to measure the particle, and he adopts a weak measurement scheme, coupling the state of the received system |ψi with the state of an ancilla system that is initially in the state|+i. Bob1’s coupling follows is given by:
|ψi ⊗ |+i −→ α |Hi ⊗ |+i + β |Vi ⊗ cos |+i + i sin |−i
We will indicate with|φHi the ancilla state associated to the|Hi polarization of the entangled particle (|φHi = |+i), and we will indicate with |φVi the ancilla state associated to the|Vi polarization (|φVi = cos |+i + i sin |−i). Bob1measures then on an orthonormal basis set|+i , |−i, obtaining the results +, − with a probability:
P(+) = P|ψi = |HiP+| |φHi
+ P|ψi = |ViP+| |φVi
(3.1) P(−) = P|ψi = |HiP−| |φHi
+ P|ψi = |ViP−| |φVi
(3.2) where P|ψi = |Hi and P|ψi = |Vi represent respectively the probability of receiving an entangled particle with polarization|Hi and |Vi, and the terms of the form P
+| |φHi
represent the conditional probability of obtaining a certain Bob1
outcome given a certain ancilla state. The outcome probabilities 3.1and3.2in the presented scheme are easy to calculate and read:
P(+) = 1 2
2 −sin2
(3.3) P(−) = 1
2sin2 (3.4)
From equations 3.4and 3.3we can easily see that both the outcome probabilities depend on the parameter by the function sin2. We can call the quantity sin2 = G information gain, we see in fact that G gives an idea of the quantity of information extracted from the entangled state by the ancilla:
• If G = 0, the read-out of the ancilla returns always the ancilla initial state, so basically the system behaves as if there was no interaction at all, and Bob1
doesn’t gain any information about the system state.
• If G = 1, the read-out of the ancilla allows to reconstruct exactly the system state, so it is like Bob1 was performing a strong measurement on the system state.
• All the values for G between 0 and 1 characterize an intermediate situation between very weak (absent) Bob1 measurement and strong measurement.
From the above consideration it is clear why we refer to G as the information gain.
Another quantity of interest is F = hφH|φVi = cos called the quality factor. The state after Bob1 measurement (defined ρ0in the density matrix notation) reads:
ρ0= F|ψi hψ| + (1 − F)
π+|ψi hψ| π++ π−|ψi hψ| π−
From such equation it is clear why we refer the to F as the quality factor: F weights the proportion of the postmeasurement state that corresponds to the original state, while the remainder corresponds to the state decohered in the measurement eigenbasis, as it would have been if measured strongly.
